On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

Details On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

2005-01-21

arxiv.org/abs/math/0501356v1

Mathematics

Combinatorics

Scientific paper

We study the multigraded Poincar\'e-Betti series of $A=S/\aaf$, where $S$ is the ring of polynomials in $n$ indeterminates divided by the monomial ideal $\aaf$. There is a conjecture about the multigraded Poincar\'e-Betti series by Charalambous and Reeves which they proved in the case, where the Taylor resolution is minimal. We introduce a conjecture about the minimal $A$-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves. We prove our conjecture for several classes of algebras. The conjecture implies that $A$ is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of $\aaf$ implies Golodness for $A$.

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